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<h4> Entering Partial Derivatives</h4>
<br>
 
<table style=width:100%>
  <tr>
    <td>Partial Derivative</td>
    <td>Enter into WeBWork</td>
  </tr>
  <tr>
    <td>\(\frac{\partial u}{\partial x}\)</td>
    <td> ux</td>
  </tr>
  <tr>
    <td>\(\frac{\partial u}{\partial t}\)</td>
    <td>ut</td>
  </tr>
  <tr>
    <td>\(\frac{\partial u}{\partial y}\)</td>
    <td>uy</td>
  </tr>
  <tr>
    <td>\(\frac{\partial^2 u}{\partial x^2}\)</td>
    <td>uxx</td>
  </tr>
  <tr>
    <td>\(\frac{\partial^2 u}{\partial t^2}\)</td>
    <td>utt</td>
  </tr>
  <tr>
    <td>\(\frac{\partial^2 u}{\partial y^2}\)</td>
    <td>uyy</tdd>
  </tr>
</table
<br>
<br>
To answer questions that require you to input a PDE you will also need to know the form
of the PDE WeBWorK is expecting. In particular you will need to use the same letters
for constants that WeBWorK is expecting. Below is a table of the PDE's 
you may encounter.
<br>
<h4> Models </h4>
<br>
<table>
  <tr>
     <td>Model</td>
     <td>PDE</td>
  </tr>
     <td>heat equation</td>
     <td>\(k \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}\)</td>
  </tr>
  <tr>
     <td>heat equation with lateral heat transfer - \(u_m\) is the temperature of the surrounding medium and will be given, h is the constant from Newton's $
     <td>\(k \frac{\partial^2 u}{\partial x^2}-h (u-um) = \frac{\partial u}{\partial t}\)</td>
  </tr>
  <tr>
     <td>wave equation</td>
     <td>\(a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}\))</td>
  </tr>
  <tr>
     <td>wave equation with damping</td>
     <td>\(a^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}+c \frac{\partial u}{\partial t}\)</td>
  </tr>
  <tr>
     <td>wave equation with an external force (f(x,t) will specified in the problem</td>
     <td>\(a^2 \frac{\partial^2 u}{\partial x^2} + f(x,t) = \frac{\partial^2 u}{\partial t^2}\)</td>
  </tr>
  <tr>
     <td>Laplace's equation</td>
     <td>\(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = 0\)</td>
  </tr>
</table>
<br>
For problems with proportional quantities the constant of proportionality is c.
<br>
<br>
<h4>  Entering Boundary and Initial Conditions </h4>
<br>
There are three types of boundary conditions we will consider:
<br>
1. Ends held at a constant temperature \(u_0\) (Dirichlet condition): \(u(L,t) = u_0\)
<br>
2. Ends insulated (Neumann condition): \(\frac{\partial u}{\partial x}\big\vert_{x=L} = 0\)
<br>
3. Heat transfer through the ends into a medium held at constant temperature \(u_m\)
<br>
<br>

Suppose that we want to enter the boundary condition \(\frac{\partial u}{\partial x}\big\vert_{x=0}=0\).
You will be given two answer blanks: the first is to input the partial derivative and the point, ux(0,t),
and the second will be for the right hand side. In WeBWorK notation the boundary condition would be given as <code>ux(0,t) = 0</code>.

